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## Homework Statement

The problem as written in Apostol:

"A farmer wishes to enclose a rectangular pasture of area A adjacent to a long stone wall. What dimensions require the least amount of fencing?"

## Homework Equations

From the problem setup, we have deduced two elementary equations:

P = 2x + y

A = xy

where P is the perimeter (or length of fence, minus the stone wall part) and A is area (a constant).

## The Attempt at a Solution

There are two equations here and two variables. That seemed like a good thing. We found y = A/x and then substituted into the perimeter equation: P(x) = 2x + A/x.

In order to optimize, we differentiated P(x):

P`(x) = 2 - (A/x^2)

Then set the equation to 0 find the extrema and to solve for the variable x.

0 = 2 - (A/x^2)

2 = A/(x^2)

x^2 = A/2

x = (A/2)^1/2

And then substituted x into the area equation to find y:

y((A/2)^1/2) = A

y = (A/(A/2)^1/2))

According to the answer key this is wrong and we've been unable to find our error. It seems like something in the setup is wrong but we've reached a block. Does anyone have any insight into this?

Thank you in advance!